Photoinduced enhancement of excitonic order in the two-orbital Hubbard model
Photoinduced dynamics in an excitonic insulator is studied theoretically by using a two-orbital Hubbard model on the square lattice where the excitonic phase in the ground state is characterized by the BCS-BEC crossover as a function of the interorbital Coulomb interaction. We consider the case where the order has a wave vector and photoexcitation is introduced by a dipole transition. Within the mean-field approximation, we show that the excitonic order can be enhanced by the photoexcitation when the system is initially in the BEC regime of the excitonic phase, whereas it is reduced if the system is initially in the BCS regime. The origin of this difference is discussed from behaviors of momentum distribution functions and momentum-dependent excitonic pair condensation. In particular, we show that the phases of the excitonic pair condensation have an important role in determining whether the excitonic order is enhanced or not.
Ultrafast control of electronic properties in materials by light irradiation has become a fascinating subject in condensed matter physics. In particular, recent experimental findings such as photoinduced localization of charges in metalsIshikawa_NatComm14 (); Kawakami_PRB17 () and appearance of novel transient orders by photoexcitationOnda_PRL08 (); Fausti_Sci11 (); Singer_PRL16 (); Mor_PRL17 () have attracted much attention. These phenomena are in sharp contrast to conventional photoinduced phase transitions typified by insulator-to-metal transitions in correlated electron systemsYonemitsu_PR08 () where photoexcitation usually melts electronic orders.
Excitonic insulators (EIs) were proposed to appear in a semimetal with a small band overlapMott_PM61 () or in a semiconductor with a small band gapKnox_SSP63 (). This state arises from macroscopic condensation of bound electron-hole pairs, excitons, that are mediated by the Coulomb interaction. Although theories of EIs have been developed since 1960sJerome_PR67 (); Kunes_JPCM15 (); Halperin_RMP68 (), its experimental identification is a challenging task. In fact, a few materials such as 1-TiSeSalvo_PRB14 (); Kidd_PRL02 (); Cercellier_PRL07 (); Monney_PRB09 (); Monney_PRL11 (); Pillo_PRB00 () and TaNiSeWakisaka_PRL09 (); Kaneko_PRB13 (); Seki_PRB14 (); Lu_NatComm17 () have been known as candidates for EIs. In this regard, a search for novel photoinduced phenomena that are peculiar to EIs is of great interest. Among the candidate materials, 1-TiSe is a semimetallic material that exhibits a charge-density-wave (CDW) state at low temperatures. Through studies of photoinduced melting dynamics of the CDW, a signature of a possible excitonic order in the CDW, i.e., an excitonic CDW has been argued from a nonequilibrium point of viewRohwer_NAT11 (); Vorobeva_PRL11 (); Porer_NatMat14 (); Monney_PRB16 (); Mathias_NatComm16 (). TaNiSe is a semiconductor with a small direct gap above K where a semiconductor-to-insulator transition occurs. It has been shown that various experimental results are consistent with the realization of an excitonic order for Lu_NatComm17 (). Quite recently, photoinduced enhancement of the gap associated with the excitonic order has been reported for this materialMor_PRL17 ().
Theoretically, most studies on EIs have been concerned with their equilibrium propertiesKaneko_PRB13 (); Phan_PRB10 (); Zocher_PRB11 (); Seki_PRB11 (); Zenker_PRB12 (); Kaneko_PRB12 (); Seki_PRB14 (); Watanabe_JPCS15 (); Nasu_PRB16 (); Tatsuno_JPSJ16 (). For instance, the BCS-BEC crossover that is an important concept for characterizing the nature of the pair condensation has been examined in EIsSeki_PRB11 (); Phan_PRB10 (); Watanabe_JPCS15 (); Kaneko_PRB12 (). In nonequilibrium conditions, photoinduced melting dynamics of the excitonic order which is accompanied by a CDW has been investigated in semimetallic systemsGolez_PRB16 (). For a direct gap semiconductor like TaNiSe, Murakami et al. have shown that photoexcitation enhances the gap by electron-phonon couplingsMurakami_arX17 (). In spite of these studies, our understandings of photoinduced properties of EIs are still limited. For example, a possibility for the gap enhancement in purely electronic models has not been elucidated so far. A relevance of the BCS-BEC crossover to photoinduced states has not been fully explored yet.
In this paper, we investigate photoinduced dynamics of EIs using a two-orbital Hubbard model on the square lattice, where we consider electric dipole transitions caused by photoexcitation. By computing momentum distribution functions and excitonic pair condensation in momentum space, we show that when the system initially possesses a BEC-type (BCS-type) excitonic order, its dynamics is essentially described by a real (momentum) space picture. In this sense, the photoinduced dynamics is strongly affected by where the system is located in the BCS-BEC crossover in thermal equilibrium. For the BEC-type order, an enhancement of the gap is realized, whereas the gap is reduced for the BCS-type order. We explain this difference by analyzing time evolution of the phases of the excitonic pair condensation.
Ii Model and Method
We consider a two-orbital Hubbard model on the square lattice defined by the following Hamiltonian,
where () is the creation operator for an electron with spin ( at the -th site on the orbital. We define and . The parameter is the transfer integral for electrons on the orbital and denotes a pair of nearest-neighbor sites. In this paper, we set and and choose the former as the unit of energy. The quantity is a parameter that controls the overlap between the and bands. The intraorbital and interorbital Coulomb interactions are denoted by and , respectively. We fix the electron density per site at .
where with being the Heaviside step function. The pulse width is denoted by and we use single cycle pulses () throughout the paper.
We apply the Hartree-Fock (HF) approximation to Eq. (1). For , the band structure corresponds to a direct gap semiconductor for . For , the Fermi surface has electron and hole pockets that coincide with each other owing to the particle-hole symmetry. The system has an instability toward excitonic condensation with a wave vector so that we define the excitonic order parameter as , which is independent of . We assume that does not depend on . Similarly, we write where is the electron density on the orbital per site. Since , we have . The total Hamiltonian within the HF approximation is written in momentum representation as
where and is the matrix defined as
In Eq. (5), we define and where and are the noninteracting energy dispersions for the and bands, respectively. In the ground state (), and are determined self-consistently.
Within the Hartree-Fock approximation, the one-particle state at time with wave vector and spin is written as
with . If is an occupied state at , the momentum distribution function for the orbital, , and the electron-hole pair condensation in -space, , are written as
In terms of these quantities, and are given as
The photoinduced dynamics is obtained by solving the time-dependent Schrdinger equation
The exponential operator is expanded with time slice until the norm of the wave function becomes unity with sufficient accuracy. For later convenience, we define the time average of a quantity as
where we use and throughout the study. We use , , and unless otherwise noted. The total number of sites is .
iii.1 Ground-state properties
Before discussing photoinduced dynamics, we present ground-state properties. In Fig. 1(a), we show the dependence of the excitonic order parameter and the charge gap , where is taken to be real. A similar plot for the electron densities and is shown in Fig. 1(b). For , the system is a metal with electron and hole pockets in the Fermi surface. Because of the perfect nesting of the Fermi surface, an infinitesimally small induces the excitonic orderWatanabe_JPCS15 (); Kaneko_PRB12 (), which results in . With increasing , exhibits a peak near and then decreases. At , a phase transition from the EI to a band insulator (BI) occurs. The density () monotonically decreases (increases) with increasing because of the Hartree shiftKaneko_PRB12 (). In the BI phase, the and bands are completely decoupled so that we have and (). These results are qualitatively consistent with previous studies for ground states that take account of electron correlationsWatanabe_JPCS15 (); Kaneko_PRB12 (), where the properties of the excitonic order are discussed in the context of the BCS-BEC crossoverSeki_PRB11 (); Phan_PRB10 (); Watanabe_JPCS15 (); Kaneko_PRB12 (). When is small, the coherence length of electron-hole pairs is large and the order is well described by the BCS theory. On the other hand, in the region near the phase boundary between the EI and the BI, is small so that the BEC picture is more appropriate for describing the EI.
iii.2 Photoinduced dynamics
In Figs. 2(a) and 2(b), we show the time evolution of for and where the system is initially in the BCS and the BEC regimes, respectively. In Figs. 2(c) and 2(d), we depict the time profile of . We use , although our qualitative results are unaltered even if we use . For , after the photoexcitation is smaller than that in the ground state, . In particular, it almost vanishes for . On the other hand, for , becomes larger than as we increase . After the photoexcitation, there is a characteristic oscillation in for both and . To analyze this oscillation, we fit a function,
to the time profile of except for the cases where after the photoexcitation. We use the time range of to determine the parameters , , , , and . The results are shown in Fig. 2, indicating that they fit well to the curves. In Fig. 3, we show the relation between and that is the time average of the transient gap . It is apparent that the frequency corresponds to , which demonstrates that the oscillation in is the Higgs amplitude modeVolkov_JETP74 (); Littlewood_PRL81 (); Pekker_ARCMP15 (). As shown in Figs. 2(c) and 2(d), is conserved for . In the case of , the behavior of is similar to that of the time-averaged . When the time-averaged is largely (slightly) enhanced after the photoexcitation, also increases largely (slightly) after that. In this case, the enhancement of is interpreted as a consequence of the Hartree shiftMurakami_arX17 (). The increase in by the photoexcitation makes the two bands and approach each other, which promotes the mixing of these bands. However, for , the behavior of is qualitatively different from that of . For example, the value of after the photoexcitation with and that with are largely different although those of are slightly different. This indicates that the photoinduced change in is not simply explained by the Hartree shift. To understand why is enhanced (suppressed) in the BEC (BCS) regime more adequately, it is important to examine the momentum distribution function and the phase of the electron-hole pair condensation in -space, which will be discussed in Sect. III. C.
In Fig. 4, we compare the energy dispersion of the bands in the ground state and the time-averaged energy levels after the photoexcitation. The latter is denoted by with being the band index. The quantity is obtained from the time average of , which is the eigenvalue of . In Fig. 4(a), we show the ground-state dispersion and with for the case of . An enlarged view near the initial gap is shown in Fig. 4(c), indicating that the initial gap disappears in since we have after the photoexcitation. Such a photoinduced gap closing has been reported recently in a system with an excitonic CDW by using the GW method that takes account of correlation effects beyond the mean-field theoryGolez_PRB16 (). Away from the gap, and are very close to each other because the change in by the photoexcitation is small as shown in Fig. 2(c). Note that the difference between and near the gap originates from the change in , while that away from the gap comes from the change in (). For the values of where is nonzero after the photoexcitation (e.g. ), the gap remains in (not shown). For , the charge gap is markedly enlarged because of the increase in . Away from , the upper and lower bands slightly approach to each other because the difference between the Hartree shifts for the two bands is reduced. We note that after the photoexcitation has only a weak dependence so that and show similar dependences even quantitatively. This is because and are conserved and the dependence comes only through whose oscillation just affects energy levels in the vicinity of the gap.
Figure 5(a) shows on the plane. The excitonic order is largely enhanced around . The enhancement occurs in a region where the initial state is in the BEC regime inside the EI phase and in the nearby BI phase where . On the other hand, when is small and the system is initially in the BCS regime, is decreased by photoexcitation. In Fig. 5(b), we show whose enhancement is most prominent near and . For , the enhancement is less clear compared to that of . This is because the initial gap in the BI phase rapidly increases with . In Fig. 5(c), we show for the case of , which indicates that the results are qualitatively the same as that for . However, a quantitative difference appears depending on the sign of , the reason of which will be discussed in Sect. III. C.
In Fig. 6, we show on the plane with for and . Compared to the case with , the region where is enhanced shifts toward larger values of and . In particular, there is almost no enhancement near for . This comes from the fact that, when is much larger than the initial gap , the charge transfer from the lower band to the upper band by the photoexcitation becomes ineffective so that the mixing of the two bands is hardly promoted. In Fig. 7, we show the time profile of and that of with and for where the initial gap is . Changes in and by the photoexcitation are small compared to those for shown in Figs. 2(b) and 2(d). shows a characteristic oscillation with a frequency corresponding to the Higgs amplitude mode as discussed above. After the photoexcitation, it becomes slightly smaller than . Recently, Murakami et al. have shown that photoinduced enhancement of the excitonic order appears in a one-dimensional spinless fermion model with electron-phonon couplingsMurakami_arX17 (). They considered mainly the BEC-type excitonic order and used the external laser field with a frequency much larger than the initial gap. Without the electron-phonon couplings, they did not find any enhancement of the order, which is consistent with our results for large . However, when the frequency is comparable to the initial gap, our results indicate that the excitonic order can be enhanced even in purely electronic systems, as shown in Fig. 5.
iii.3 Origin of photoinduced enhancement or suppression of
Here, we discuss the origin of distinctive dynamics induced by the dipole transitions in the BCS and BEC regimes. In Figs. 8(a) and 8(b) [9(a) and 9(b)], we show and , respectively, for () with different values of . In the ground state, for exhibits a steep change along the lines from to and from to , reflecting the energy dispersion shown in Fig. 4(a). In Figs. 8(c) and 8(d), we show enlarged views of Figs. 8(a) and 8(b) near , respectively, where we define at which holds. At , is sharply peaked at the maximum. We note that, because of Eqs. (7) and (8), has its maximum value of 0.5 when . This relation holds even at . The abrupt change in and of the ground state in -space indicates the BCS nature of the EI. When is nonzero, is strongly affected near . In particular, decreases for , whereas it increases for . This means that, for (), -electrons (-electrons) are mainly transferred to the upper band by the photoexcitation. This characteristic dependence of cannot be explained by the Hartree shift. After the photoexcitation of , has three peaks near which we label as A, B, and C in Fig. 8(d). At these points, we have and . Both and show abrupt changes in -space indicating that the excitonic order still has the BCS nature even after the photoexcitation. For and , is enhanced around , however becomes smaller than , as shown in Fig. 2(a). This indicates that the photoinduced changes in the phases of depend strongly on in the Brillouin zone. For , and of the ground state gradually vary with , as shown in Figs. 9(a) and 9(b), respectively, because of the BEC nature of the EI. In contrast to the case of , is increased by the photoexcitation for all . In Figs. 9(c) and 9(d), we show and from to . Although decreases around , it increases in a large area of the Brillouin zone. The BEC nature of the excitonic order is maintained through the photoexcitation since and gradually change with as in the ground state.
In Figs. 10(a) and 10(b), we show the time profile of the phase of for and in the case of . For comparison, we also show the phase of () in the right panels. We recall that and are related by Eq. (10). In Figs. 10(c) and 10(d), we show an enlarged view of Figs. 10(a) and 10(b), respectively, as in Fig. 4. For , the time profile of depends strongly on in the region where and show abrupt changes. Away from this region, their time profiles become in phase with that of . On the other hand, for , shows only a weak dependence except for the region near the point, and their time profiles are in phase with that of in a wide area of the Brillouin zone. In particular, at the peak position of is in phase with that of as shown in Fig. 10(d), which is in sharp contrast to the case of . The behavior of for is analyzed by using the equation of motion for written as
After the photoexcitation, and the second term on the right-hand side of Eq. (15) becomes very small at A, B, and C in Fig. 8(c) because of the relation . Therefore, at these points, essentially determines the time evolution of and the period of should be given by . In Fig. 11(a), we show and near after the photoexcitation, whereas the time evolution of at A, B, and C is depicted in Fig. 11(b). We have , , and at A, B, and C, reflecting , , and as shown in Fig. 11(a), respectively. Moreover, at A and C, the period of coincides with estimated from and , demonstrating that the results are consistent with the above arguments. These arguments show that for the region where is large, the time profiles of the phases of are out of phase for , which inhibits the enhancement of . For , since the time profile of is in phase with that of , the increase in in the large area of the Brillouin zone gives rise to the enhancement of . In this case, the dynamics of the order parameter is well described by the real space picture.
With these results, we discuss the detailed structure of Fig. 5. When is small and the initial EI state is in the BCS regime, is basically suppressed by the photoexcitation regardless of the values of . This is because the time profile of strongly depends on especially near the peak positions of as shown in Figs. 10(a) and 10(c). When is large and the initial EI state is in the BEC regime, is slightly reduced by the photoexcitation for small . In order to explain the reason for this, we first discuss the effect of the sign of on our results. At , we choose as real and positive. In this case, the magnitude of the off-diagonal elements of initially increases if , whereas it decreases if . This initial increase (decrease) results in an enhancement (suppression) of just after the external field is switched on, as observed for the case of in Figs. 2(a) and 2(b). We show the time profile of and for with in Fig. 12, which in fact indicates that initially decreases when is switched on. By this effect, the region where is enhanced for is narrower than that for as shown in Figs. 5(a) and 5(c). For small with (), the initial reduction of dominates the behavior of . For , the charge transfer from the lower band to the upper band dominates over this effect, so that is enhanced even in the case of . For small () with , is slightly suppressed for large since becomes smaller than that in the ground state around the point as shown in Fig. 9(d). We note that the time profile of slightly deviates from that of in this region, whereas it is almost in phase with that of away from the point.
When is slightly smaller than , is suppressed compared to in a region of large (), although it is enhanced around . In Fig. 13(a) [13(b)], we show () for and where we have . Between and , has two large peaks as shown in Fig. 13(d), which we label as D and E for the case of . The peak at D appears since crosses 0.5, whereas E reflects a sharp peak in . Although is enhanced near the point E, their values away from the two peaks are slightly smaller than those in the ground state. In Fig. 14, we show the time profile of from to for . Notably, strongly depends on in a region including D and E. The time profile of at E is quite different from that of . Although the time profile of away from this region is in phase with that of , the values of are smaller than those in the ground state, so that they do not contribute to enhance . These results indicate that even when the initial state is the BEC-type EI, whether is enhanced or not depends on the value of .
In this paper, we study dipole-transition-induced dynamics of excitonic orders by using the two-orbital Hubbard model on the square lattice. We show that the photoinduced dynamics depends strongly on whether the EI is initially in the BCS regime or in the BEC regime. The excitonic order is basically enhanced by the photoexcitation in the latter, whereas it is reduced in the former. These results are caused by different behaviors of the momentum distribution functions and the phases of the electron-hole pair condensation in -space. When the initial EI is of the BEC-type, its dynamics is interpreted from the real space picture, whereas the dependence of physical quantities is essential for the BCS-type EI.
Acknowledgements.The authors thank Y. Murakami for fruitful discussions. This work was supported by Grants-in-Aid for Scientific Research (C) (Grant No. 16K05459) and Scientific Research (A) (Grant No. 15H02100) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
- (1) T. Ishikawa, Y. Sagae, Y. Naitoh, Y. Kawakami, H. Itoh, K. Yamamoto, K. Yakushi, H. Kishida, T. Sasaki, S. Ishihara, Y. Tanaka, K. Yonemitsu, and S. Iwai, Nat. Commun. 5, 5528 (2014).
- (2) Y. Kawakami, Y. Yoneyama, T. Amano, H. Itoh, K. Yamamoto, Y. Nakamura, H. Kishida, T. Sasaki, S. Ishihara, Y. Tanaka, K. Yonemitsu, and S. Iwai, Phys. Rev. B 95, 201105(R) (2017).
- (3) K. Onda, S. Ogihara, K. Yonemitsu, N. Maeshima, T. Ishikawa, Y. Okimoto, X. Shao, Y. Nakano, H. Yamochi, G. Saito, and S. Koshihara, Phys. Rev. Lett. 101, 067403 (2008).
- (4) D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri, Science 331, 189 (2011).
- (5) A. Singer, S. K. K. Patel, R. Kukureja, V. Uhli, J. Wingert, S. Festersen, D. Zhu, J. M. Glownia, H. T. Lemke, S. Nelson, M. Kozina, K. Rossnagel, M. Bauer, B. M. Murphy, O. M. Magnussen, E. E. Fullerton, and O. G. Shpyrko, Phys. Rev. Lett. 117, 056401 (2016).
- (6) S. Mor, M. Herzog, D. Golez, P. Werner, M. Eckstein, N. Katayama, M. Nohara, H. Takagi, T. Mizokawa, C. Monney, and J. Stahler, Phys. Rev. Lett. 119, 086401 (2017).
- (7) K. Yonemitsu and K. Nasu, Phys. Rep. 465, 1 (2008).
- (8) N. F. Mott, Phil. Mag. 6, 287 (1961).
- (9) R. S. Knox, Solid State Phys. Suppl. 5, 100 (1963).
- (10) D. Jerome, T. M. Rice, and W. Kohn, Phys. Rev. 158, 462 (1967).
- (11) B. I. Halperin and T. M. Rice, Rev. Mod. Phys. 40, 755 (1968).
- (12) J. Kunes, J. Phys.: Cond. Mat. 27, 333201 (2015).
- (13) F. J. Di Salvo, D. E. Moncton, and J. V. Waszczak, Phys. Rev. B 14, 4321 (1976).
- (14) Th. Pillo, J. Hayoz, H. Berger, F. Lvy, L. Schlapbach, and P. Aebi, Phys. Rev. B 61, 16213 (2000).
- (15) T. E. Kidd, T. Miller, M. Y. Chou, and T.-C. Chiang, Phys. Rev. Lett. 88, 226402 (2002).
- (16) H. Cercellier, C. Monney, F. Clerc, C. Battaglia, L. Despont, M. G. Garnier, H. Beck, P. Aebi, L. Patthey, H. Berger, and L. Forr, Phys. Rev. Lett. 99, 146403 (2007).
- (17) C. Monney, H. Cercellier, F. Clerc, C. Battaglia, E. F. Schwier, C. Didiot, M. G. Garnier, H. Beck, P. Aebi, H. Berger, L. ForrÃ³, and L. Patthey, Phys. Rev. B 79, 045116 (2009).
- (18) C. Monney, C. Battaglia, H. Cercellier, P. Aebi, and H. Beck, Phys. Rev. Lett. 106, 106404 (2011).
- (19) Y. Wakisaka, T. Sudayama, K. Takubo, T. Mizokawa, M. Arita, H. Namatame, M. Taniguchi, N. Katayama, M. Nohara, and H. Takagi, Phys. Rev. Lett. 103, 026402 (2009).
- (20) T. Kaneko, T. Toriyama, T. Konishi, and Y. Ohta, Phys. Rev. B 87, 035121 (2013).
- (21) K. Seki, Y. Wakisaka, T. Kaneko, T. Toriyama, T. Konishi, T. Sudayama, N. L. Saini, M. Arita, H. Namatame, M. Taniguchi, N. Katayama, M. Nohara, H. Takagi, T. Mizokawa, and Y. Ohta, Phys. Rev. B 90, 155116 (2014).
- (22) Y. F. Lu, H. Kono, T. I. Larkin, A. W. Rost, T. Takayama, A, V. Boris, B. Keimer, and H. Takagi, Nat. Commun. 8, 14408 (2017).
- (23) T. Rohwer, S. Hellmann, M. Wiesenmayer, C. Sohrt, A. Stange, B. Slomski, A. Carr, Y. Liu, L. M. Avila, M. Kallane, S. Mathias, L. Kipp, K. Rossnagel, and M. Bauer, Nature 471, 471 (2011).
- (24) E. Mhr-Vorobeva, S. L. Johnson, P. Beaud, U. Staub, R. De Souza, C. Milne, G. Ingold, J. Demsar, H. Schaefer, and A. Titov, Phys. Rev. Lett. 107, 036403 (2011).
- (25) M. Porer, U. Leierseder, J.-M. Mnard, H. Dachraoui, L. Mouchliadis, I. E. Perakis, U. Heinzmann, J. Demsar, K. Rossnagel, and R. Huber, Nat. Mat. 13, 857 (2014).
- (26) C. Monney, M. Puppin, C. W. Nicholson, M. Hoesch, R. T. Chapman, E. Springate, H. Berger, A. Magrez, C. Cacho, R. Ernstorfer, M. Wolf, Phys. Rev. B 94, 165165 (2016).
- (27) S. Mathias, S. Eich, J. Urbancic, S. Michael, A. V. Carr, S. Emmerich, A. Stange, T. Popmintchev, T. Rohwer, M. Wiesenmayer, A. Ruffing, S. Jakobs, S. Hellmann, P. Matyba, C. Chen, L. Kipp, M. Bauer, H. C. Kapteyn, H. C. Schneider, K. Rossnagel, M. M. Murnane, and M. Aeschlimann, Nat. Commun. 7, 12902 (2016).
- (28) V. -N. Phan, K. W. Becker, and H. Fehske, Phys. Rev. B 81, 205117 (2010).
- (29) B. Zocher, C. Timm, and P. M. R. Brydon, Phys. Rev. B 84, 144425 (2011).
- (30) K. Seki, R. Eder, and Y. Ohta, Phys. Rev. B 84, 245106 (2011).
- (31) B. Zenker, D. Ihle, F. X. Bronold, and H. Fehske, Phys. Rev. B 85, 121102(R) (2012).
- (32) T. Kaneko, K. Seki, and Y. Ohta, Phys. Rev. B 85, 165135 (2012).
- (33) H. Watanabe, K. Seki, and S. Yunoki, J. Phys.: Conf. Ser. 592, 012097 (2015).
- (34) J. Nasu, T. Watanabe, M. Naka, and S. Ishihara, Phys. Rev. B 93, 205136 (2016).
- (35) T. Tatsuno, E. Mizoguchi, J. Nasu, M. Naka, and S. Ishihara, J. Phys. Soc. Jpn. 85, 083706 (2016).
- (36) D. Gole, P. Werner, and M. Eckstein, Phys. Rev. B 94, 035121 (2016).
- (37) Y. Murakami, D. Gole, M. Eckstein, P. Werner, Phys. Rev. Lett. 119, 247601 (2017).
- (38) A. Terai and Y. Ono, Prog. Theor. Phys. Suppl. 113, 177 (1993).
- (39) M. Kuwabara and Y. Ono, J. Phys. Soc. Jpn. 64, 2106 (1995).
- (40) Y. Tanaka and K. Yonemitsu, J. Phys. Soc. Jpn. 79, 024712 (2010).
- (41) A.F. Volkov, Sh. M. Kogan, Sov. Phys. JETP 38, 1018 (1974).
- (42) P. B. Littlewood and C. M. Varma, Phys. Rev. Lett. 47, 811 (1981).
- (43) D. Pekker and C. M. Varma, Annu. Rev. Condens. Matter Phys. 6, 269 (2015).